Ionic electrodiffusion is important to many fields, including biology, chemistry and physics, and has wide applications [1]}. There are extensive mathematical studies of models coupling the Nernst-Planck equations with various fluid dynamical systems. For the fluids that are described by the Navier-Stokes equations, the system is known as the Nernst-Planck-Navier-Stokes (NPNS) system. In the whole space \(\mathbb {R}^d\)  (\(d = 2, 3\) ), local existence of solutions of the NPNS system is obtained in [2]}, and later weak solutions are proved to exist globally in time [3]}, with some \(L^2\)  decay if the dimension is two [4]}. In bounded domains \(\Omega \subset \mathbb {R}^d\) , the NPNS system has global solutions under various appropriate boundary conditions. For example, with blocking boundary conditions, global weak solutions exist in both two and three dimensions [5]}, [6]}. For ionic concentrations satisfying blocking boundary conditions while the electrical potential satisfying the Dirichlet boundary condition, global weak solutions exist in two and three dimensions if the initial data is small [7]}. In the case of blocking boundary conditions for the ionic concentrations and homogeneous Neumann boundary condition for the electrical potential, weak solutions are global in two dimensions [8]}. Moreover, with blocking boundary conditions for the ionic concentrations and Robin boundary condition for the electrical potential, two dimensional strong solutions are global [9]}. The same result holds in three dimensions if the fluid velocity remains regular for all time [10]}. When the ionic concentrations satisfy either the blocking boundary conditions or the uniformly selective boundary conditions, strong solutions are global in two dimensions [11]}, and in three dimensions provided that the initial data is a small perturbation of a steady state [12]}. If both the ionic concentrations and the electrical potential obey the Dirichlet boundary conditions, global strong solutions exist in three dimensions as long as the fluid velocity is regular [13]}. With periodic boundary conditions, two dimensional strong solutions exist globally in time, and long time behaviors of the solutions are studied in [14]} under the influence of body forces or body charges. Some zero Debye length limit (\(\varepsilon \rightarrow 0\)  in ()) results for the NPNS system are proved in [15]}, [16]}, [17]}, [18]}. In the limit of zero viscosity in the Navier-Stokes equations, the solutions of NPNS system in two dimensions converges to the solutions of the corresponding Nernst-Planck-Euler (NPE) system, whose solutions exist and are global [19]}, [20]}, [21]}. For the Nernst-Planck system coupled with time dependent Stokes equations, solutions are known existing globally in time in three dimensions [13]}, [10]}.
